Transcript Document

Understanding
weak interactions
“Symmetry E70
(Butterflies)”
by M.C. Escher - 1948
“Symmetry E72
(Fish and Boats)”
by M.C. Escher - 1949
Understanding the origins and magnitudes of
weak interactions
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Molecular crystals being made up of molecules are affected
by intramolecular and intermolecular forces:
• Intramolecular forces affect the physical properties and shape
of molecules which are important in crystal packing
• The intermolecular forces are generally much weaker and
short-range in their effect
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This combination of strong and weak introduces diversity in
the properties of molecular crystals
This is in contrast to simple ionic crystals which are
dominated by strong long-range columbic forces
Understanding the origins and magnitudes of
weak interactions contd.
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Understanding the origins and magnitudes of intermolecular
forces is therefore necessary to understanding the
properties of molecular crystals
In particular it is necessary to understand their dependence
on the following:
• molecular properties
• intermolecular separation
• intermolecular orientation
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The understanding of intermolecular interactions in the
context of crystal packing and the utilization of such
understanding in the design of new solids with desired
physical and chemical properties is in fact the focus of two
closely related fields: crystal engineering and
supramolecular chemistry
Crystal Engineering
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The field of crystal engineering aims to predict and control
crystal assembly, and hence structure, solid state
properties and reactivity.
Has a role to play in many fields: materials design, binding
of dyes onto clothing, the understanding of bone growth,
the formation of clatharate hydrates which block pipes in
the oil industry, melting point suppression, etc.
Has found application in the field of “green” chemistry
where reactions are carried out without the use of solvents,
and where the reactions observed often do not have
solution phase counterparts.
Some examples of weak interactions in action
Van Der Waals Forces
Dipole-dipole
Dipole-induced dipole
Dispersion/London forces
All interactions reflect a balance between attractive and repulsive forces
N
I
Interaction between dipolar molecules
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The electric field produced by a dipole μ along its own
direction at a distance r from its centre is 2μ/r3
For two dipoles aligned head to tail at a distance r apart,
the interaction energy U between them is given by:
U = -2μ1μ2/r3
Interaction between dipolar molecules
contd.
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However, in crystals the dipoles will not be necessarily be
well aligned with each other.
By considering:
• the random orientation of the dipoles,
• defining their relative orientations using polar coordinates,
• and including attractive and repulsive interaction components,
the following equation may be obtained:
U = -(μ1μ2/r3){2cos θ1cos θ
2
- sin θ1sin θ2cos(φ1 - φ2)}
Attractive component
Repulsive component
Interaction between dipolar molecules
contd.
U = -(μ1μ2/r3){2cos θ1cos θ
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2
- sin θ1sin θ2cos(φ1 - φ2)}
The above equation shows that the interaction energy
between two dipoles in a crystal will be inversely
proportional to r3
The equivalent equation for an interaction between two
dipoles in solution has a dependence proportional to r6
The dipole-dipole interaction in the solid-state therefore has
a longer effective range
Lastly, and most importantly the above equation has
repulsive and attractive components so depending on the
relative orientations of the dipoles this interaction can be
attractive or repulsive
Dipole-induced dipole interactions
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The electric field of one dipole can induce a dipole on a
second polarizable molecule
This interaction is dependent on the component μ1 of the
electric field of the dipole along the line joining it and the
molecule centered at distance r
It is also dependent on the polarizability α2 of the second
molecule
The interaction between the dipole and induced dipole is
given by the following equation:
U = -4α2μ12/r6
Note the dependence proportional to r6 indicating that this
is a very short range interaction
Dipole-induced dipole interactions contd.
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In contrast to dipole-dipole interactions, this interaction is
always attractive as only the magnitude of the interaction
depends on the relative orientations of the two molecules
Since polar molecules can also be polarized, this interaction
also contributes to the total interaction energy between the
two dipolar molecules, i.e.
Utotal = Udipole-dipole + ΣUdipole-induced dipole
Dispersion/London forces
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Non-polar molecules can interact with each other even
though they do not have permanent dipoles
A dipole in one molecule can be created from small
instantaneous and transient changes in the positions of
electrons (charge displacements = r1 = x1y1z1) in the
molecule leading to a transient dipole
This transient dipole can then induce a dipole in the second
molecule with the appropriate charge displacements
These two dipoles attract each other leading to a decrease
in the potential energy of the system
The attraction between the molecules depends strongly on
their polarizability
Dispersion/London forces contd.
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The total potential energy is therefore made up from the
energy needed to produce the dipoles (I) and the energy of
interaction between the two resulting dipoles (II):
U = e2r12/2α + e2r22/2α + (e2/r3)(x1x2 + y1y2 - 2z1z2) (1)
I
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Allowing for notation Term II is the same as the equation
for a dipole-dipole interaction that was shown earlier
U = -(μ1μ2/r3){2cos θ1cos θ
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II
2
- sin θ1sin θ2cos(φ1 - φ2)}
Note that the repulsion term is present. This is because
transient dipoles can also be repulsive. However, because
of induction the interaction is on average attractive and not
zero as one would expect. (See Atkins)
Dispersion/London forces contd.
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An alternative and more powerful form of equation (1) can
be obtained by expressing the transient dipole as an
oscillator:
E0 = 3hv0 - (3/4)hv0α2r-6 + .......
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The 3hv0 term is the zero point energy of the two isolated
species (depends on the ionization energy of each species)
while the second term is the attractive dispersion energy
v0 is the characteristic frequency of one of the species (how
often it “flickers”)
The attraction between the molecules depends strongly on
their polarizability
(2)
Dispersion/London forces contd.
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In practice, more complex charge displacements can occur
between molecules leading to quadrupoles and higher
multipoles
The true dispersion force experienced by a molecule is
therefore better described by the following equation:
Udispersion = c6r-6 + c8r-8 + c10r-10 + ........
where c6, c8 etc. are constants.
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Quadropoles and higher terms have often been neglected
but can be very important. In CO2 the heat of sublimation
at 0 K (27 kJ/mol) has been found to be 45% due to
quadropole-quadropole interactions and 55% due to simple
dispersion interactions.
Final comments on VDW forces
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Dipole-dipole, dipole-induced dipole and dispersion forces
are often collectively referred to as van der Waals forces
The above expressions show that VDW forces should be
largest between polar or very polarizable molecules
In general polarizability is known to increase with increased
molecular volume (larger molecules) and increased
numbers of π bonds
Comparison of interaction energies (Atkins – 8th edition)
Interaction type Distance dependence Typical energy/(kJ mol-1)
Ion-ion
1/r
250
Ion-dipole
1/r2
15
Dipole-dipole
1/r3
2
Dispersion
1/r6
2
H-bond
NA
20
Repulsive forces
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The closest distance molecules can reach each other is the
point where attractive and repulsive forces are exactly in
balance
The repulsive forces start becoming significant when the
electron clouds of two molecules begin to overlap
There are two sources for this force:
• The Pauli exclusion principle – maximum of 2 electrons per
orbital with opposite spins
• Decreased electron density in areas where overlap has
occurred hence less effective shielding and hence greater
coulombic repulsion between the nuclei on the two molecules
Repulsive forces contd.
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Repulsive forces are very difficult to calculate since they
depend on the shapes and nature of the molecule of
interest
Instead, empirical potentials are used and these tend to
have the form
ar-n (n equal to 12 usually) or be-cr
with a, b, c and n being empirically determined constants
for individual atoms types
Since repulsion forces are inversely dependent on r12, they
are only important at very close range
Also, the above equations assume isotropic repulsions
between atoms. In practice anisotropic versions of these
equations are used as these better describe the behavior of
real crystal structures.
Summing attractive and repulsive components together –
atom-atom potentials
U = (qiqj)/(Drij) + A/rij12 - C/rij6
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Where qi and qj are the fractional charges on the atoms,
D is the effective dielectric constant,
Akl is the repulsive coefficient,
and Ckl the attractive coefficient.
If one removes the electrostatic
component (to make the above
equation more general) then one gets
left with the 6:n or Lennard-Jones’
form:
Ukl(rij) = Akl/rij12 - Ckl/rij6
Some typical atom-atom potentials as implemented in a
typical lattice energy program
What about H-bonding?
U = (qiqj)/(Drij) + A/rij12 - C/rij6
1 kcal = 4.184 kJ
H-bonding contd.
H-bonding contd.
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Strong H-bonds (ionic hydrogen bonds) are formed by
groups containing an electron density deficiency on the
donor group, i.e. OH+, NH+, or an excess of electron density
in the acceptor group, i.e., F-, OH-, C-O-, P-O- etc.
Moderate H-bonds are generally formed by neutral donor
and acceptor groups, i.e., OH, NH, and –O-, C=O, NAr in
which donor atoms are electronegative to the H-atom and
acceptor atoms have unshared lone-pair electrons.
O
H
O
H
H
O
O
H
H
O
O
H
H
O
O
H
H
O
O
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O
CH3
O
Weak H-bonds are formed when
the H-atom b)is covalentlyc)
a)
bonded to a slightly more electronegative atom, as in C-H,
Si-H, or when the acceptor has no lone pairs but has π
electrons.
H-bonding contd.
H-bonding contd.
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H-bonds have group-pair properties, e.g., P-OH, H-O-H and
C-OH have different donor and acceptor abilities resulting in
different H-bonds with different D…A distances and angles.
Covalent interactions on the other hand are more atomatom pair in nature. This means that C-C, C=C, C-N etc
bonds can be more easily classified into typical bond
lengths and angles, VDW and covalent radii etc…
Type
Functional group involved
Reliable donor
-OH, -NH2, -NHR, -CONH2, -CONHR, -COOH
Occasional donor -COH, -XH, -SH, -CH
Reliable acceptor -COOH, -CONHCO-, -NHCONH-, -CON<,
>P=O, >S=O, -OH
Occasional
acceptor
>O, -NO2, -CN, -CO, -COOR, -N<, -Cl
Three- and four-centre H-bonding
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One of the consequences of H-bonding being mostly
electrostatic in nature is its ability to form bifurcated (3
centre) and trifurcated (4 centre) bonds, i.e., it is possible
for an H-atom to interact with more than one acceptor.
These commonly occur when the number of acceptors
exceed the numbers of donors in a crystal, e.g. in
carbohydrates, nucleosides and nucleotides where a lot of
ether (R-O-R) or carbonyl groups are present.
A
A
A
D H
+
H N H
A
A
+
H N H
H
H
A
A
A
+
A
A
H N H
A
H
A
A
A
2- and 3-centre H-bonds typically found in amino acid structures
Three- and four-centre H-bonding contd.
What are the criteria for picking out trifurcated H-bonds?
1.
Since the interaction is an attractive one the H atom
should lie with 0.2 Å from the plane defined by the donor
and two acceptor atoms.
2.
Alternatively, the 3 angles around the H atom,
a + b + c ≈ 360°.
A
a
D H
c
b
A
Resonance assisted hydrogen bonding (RAHB)
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H-bonding can often be assisted by the presence of
conjugated π bonds. In effect the π bonds act to
cooperatively assist the H-bond making the interaction
more covalent in character and as a consequence stronger.
This phenomenon is referred to as Resonance-Assisted
Hydrogen bonding (RAHB).
H
O
d1
O
d4
d4
d2
O
H
d3
d3
d1
d2
O
H
O
O
O
O
H
Resonance assisted hydrogen bonding (RAHB) contd.
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H
O
d1
O
d4
The effect of RAHB on H-bonding has been illustrated by
examining molecules of the following type, in which the
degree of delocalization/conjugation (Q) was determined
from the molecular bond lengths, and plotted against the
H-bond O…O distance. Increased delocalization (lower Q
values) leads to shorter O…O lengths.
d4
d2
O
H
d3
d3
d1
d2
O
H
O
O
O
O
Q = d1 – d4 + d3 – d2
H
Principles of Crystal Packing
Trends Followed by Molecules when
Forming a Crystal
Rule 1
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Maximize density and minimize free volume
This is the primary packing rule in all kinds of crystals and
is referred to as Kitaigorodsky’s principle of close packing
It is especially important in polymorphic systems for
crystals that are stable at T = 0 K where no energy is
available to support more open structures
At higher temperatures, entropy (TS) plays a large role in
allowing polymorphs with more molecular freedom to occur
Rule 2
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Satisfy hydrogen bond donors and acceptors
and any other special kind of intermolecular
interactions
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H-bonds are very strong in comparison with VDW forces. As
a consequence their presence is structure determining in
molecular crystals.
Also, in a structure, the strongest hydrogen bond donors
always connect to the strongest hydrogen acceptors
Rule 3
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Minimize electrostatic energy
Like - like repulsive interactions must be minimized in favor
of like - unlike attractive interactions
Lets look now at the observed space
group frequencies
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Though there are 230 space groups more than 75% of all
organic molecules crystallize in only 10 of these –
probably 99% of all molecules crystallize in just 30 space
groups
The top ten are: P21/c, P-1, P212121, P21, C2/c, Pbca,
Pnma, Pna21, Pbcn and P1
Now that we have some understanding of the nature of
weak interactions can we rationalize why these space
groups are so popular?
• Are some symmetries more favorable than others?
Principles of Crystal Packing
Classification of Symmetries into
Favorable and Unfavorable
Inversion centers are favorable
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In centrosymmetric space groups molecules are related to
each across a centre of symmetry
Inversion centers are especially favorable for crystal
packing since they diminish like-like interactions and
are compatible with translation.
They are unique in that they change the direction but not
the orientation of intermolecular vectors, i.e. minimize the
repulsive component of dipole-dipole interactions
Molecules related by inversion centers are often connected
to each other by hydrogen bonding which is an energy
lowering interaction
H
O
H
H
O
O
H
H
O
O
H
H
O
H
H
O
Inversion centers are favorable contd.
Mirror planes are always occupied,
usually by mirror-symmetric molecules
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Unoccupied mirror planes are especially unfavorable
because they require like-like interactions between adjacent
molecules
This symmetry maximizes the repulsive component of
the dipole-dipole interaction
If unoccupied, they produce a sheet of empty space in
the crystal which has serious consequences in terms of
packing density.
Mirror planes are always occupied,
usually by mirror-symmetric molecules
contd.
Mirror symmetric molecules
Note relatively empty space
Groups with 3-, 4- and 6-fold rotation axes do not
usually occur unless the axes are located within
molecules of appropriate symmetry
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The reason for this is that it is difficult to fill the space
around these rotation axes unless the molecules have
the shape of the symmetry. The problem is especially acute
for 4- and 6-fold axes. How many organic molecules
containing a 4-fold axis do you know?
If unoccupied, they create an infinite rod of empty
space with a diameter of 3-3.5 Å in the crystal, hence
lowering the packing density of the structure
To compensate for this, areas where these occur are usually
filled with disordered solvent leading to the formation of a
solvate
Groups with 3-, 4- and 6-fold rotation axes contd.
Benzene
Twofold axes are sometimes occupied
and sometimes not
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There seems to be no energy gain or loss associated with
the use of this symmetry
As mentioned for rotation axes above, if they are
unoccupied they can lead to lower packing densities
Finally
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21 screw axes are more favorable than glide planes, which
are comparable to pure translations.
Translations are however still very favorable otherwise
crystals would not exist.
Translations tend to occur together with other symmetry
elements, i.e. P1 is not that popular
Having gone through the last few slides have a look
at the top 10 space groups again:
• P21/c, P-1, P212121, P21, C2/c, Pbca, Pnma, Pna21, Pbcn
and P1
Supplementary – examples of weak interactions